UNIPLANAR MOTION OF A RIGID BODY 237 (a) When MI and M2, the masses of the wheel and axle, are negligible; (b) When they are not negligible. 3. In the Atwood machine problem show that if the pulley is not rough enough the acceleration of the two moving masses is -z—~" m&** a M 4- me*** y where AC is the coefficient of friction. Hint. — If T and Tf are the tensile forces in the string on the two sides, T= TCP*. 4. Same as the third illustrative problem, but the pulley P is supposed to rotate. 5. In the preceding problem suppose the cylinder to roll up an inclined plane. 6. A tape of negligible mass and thickness is wound around the middle of a cylinder. The free end of the tape is attached to a fixed point and then the cylinder is allowed to fall. Show that the cylinder falls with an acceleration of f g and the tensile force of the tape is J TF, where W is the weight of the cylinder. 7. In the preceding problem the fixed point is on an inclined plane and the cylinder rolls down the plane. 8. Discuss the motion of a log which moves along its length down an inclined plane, upon two rollers, which stay horizontal. 9. A uniform rod is allowed to fall from a position where its lower end is in contact with a rough plane and it makes an angle a. with the horizon. Show that when it becomes horizontal its angular velocity is y — sin a, where Z is the length of the rod. 10. Discuss the motion of a cylinder down an inclined plane, supposing the contact to be imperfectly rough, so that the cylinder both slides and rolls. 11. In the preceding problem suppose the cylinder to be hollow.